Quantitative Tverberg theorems over lattices and other discrete sets

This paper presents a new variation of Tverberg's theorem. Given a discrete set $S$ of $R^d$, we study the number of points of $S$ needed to guarantee the existence of an $m$-partition of the points such that the intersection of the $m$ convex hulls of the parts contains at least $k$ points of $S$. The proofs of the main results require new quantitative versions of Helly's and Carath\'eodory's theorems.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1503.0611

Quantitative combinatorial geometry for continuous parameters

We prove variations of Carath\'eodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lov\'asz's colorful Helly theorem, B\'ar\'any's colorful Carath\'eodory's theorem, and the colorful Tverberg theorem.Comment: 22 pages. arXiv admin note: substantial text overlap with arXiv:1503.0611

Quantitative Tverberg, Helly, & Carath\'eodory theorems

This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the hypothesis and the conclusion. Typical measurements include the volume, the diameter, or the number of points in a lattice.Comment: 33 page

Process-based and correlative modeling of desert mistletoe distribution: a multiscalar approach

Because factors affecting distributional areas of species change as scale (extent and grain) changes, different environmental and biological factors must be integrated across geographic ranges at different resolutions, to understand fully the patterns and processes underlying species' ranges. We expected climate factors to be more important at coarse resolutions and biotic factors at finer resolutions. We used data on occurrence of a parasitic plant (Phoradendron californicum), restricted to parts of the Sonoran and Mojave deserts, to analyze how climate and mobility factors explain its distributional area. We developed analyses at five spatial resolutions (1, 5, 10, 20, 50 km) within the distributional area of the disperser species, and compared ecological niche models from three commonly used correlative methods with a process-based model that estimates colonization and extinction rates in a metapopulation framework. Correlative models improved when layers associated with hosts and disperser were used as predictors, in comparison with models based on climate only; however, they tended to overfit to data as more layers were added. Dispersal-related parameters were more relevant at finer resolutions (1–5 km), but importance of extinction-related parameters did not change with scale. We observed greater coincidence between correlative and process-based models when based only on dimensions of the abiotic niche (i.e., climate), but a clearer and more comprehensive mechanistic understanding was derived from the process-based algorithm

Quantitative Combinatorial Geometry for Continuous Parameters

We prove variations of Carathéodory’s, Helly’s and Tverberg’s theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lovász’s colorful Helly’s theorem, Bárány’s colorful Carathéodory’s theorem, and the colorful Tverberg’s theorem

Tverberg's theorem is 50 Years Old: A survey

This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. © 2018 American Mathematical Society

Tverberg Plus Minus

We prove a Tverberg type theorem: Given a set A Rd in general position with | A| = (r- 1) (d+ 1) + 1 and k∈ { 0 , 1 , … , r- 1 } , there is a partition of A into r sets A1, … , Ar (where | Aj| ≤ d+ 1 for each j) with the following property. There is a unique zj=1raffAj and it can be written as an affine combination of the element in Aj: z=∑x∈Ajα(x)x for every j and exactly k of the coefficients α(x) are negative. The case k= 0 is Tverberg’s classical theorem. © 2018, Springer Science+Business Media, LLC, part of Springer Nature

Cutting the same fraction of several measures

We study some measure partition problems: Cut the same positive fraction of $d+1$ measures in $\mathbb R^d$ with a hyperplane or find a convex subset of $\mathbb R^d$ on which $d+1$ given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.Comment: 7 pages 2 figure

The Big Questions For Biodiversity Informatics

This is the publisher's version, which the author has permission to share. The original version may be found at http://dx.doi.org/10.1080/14772001003739369Science is a sequence of generating new ideas, detailed explorations, incorporation of the results into a toolbox for understanding data, and turning them into useful knowledge. One recent development has been large-scale, computer-aided management of biodiversity information. This emerging field of biodiversity informatics has been growing quickly, but without overarching scientific questions to guide its development; the result has been developments that have no connection to genuine insight and forward progress. We outline what biodiversity informatics should be, a link between diverse dimensions of organismal biology – genomics, phylogenetics, taxonomy, distributional biology, ecology, interactions, and conservation status – and describe the science progress that would result. These steps will enable a transition from ‘gee-whiz’ to fundamental science infrastructure

La actividad “Taller” como herramienta para fomentar y evaluar competencias transversales: Expresión escrita y autoaprendizaje

Las competencias transversales suelen ser definidas como destrezas, aptitudes y habilidades deseables y propias de cada perfil profesional; éstas, no se imparten de forma directa dentro de la temática curricular de las asignaturas, pero sin embargo deben ser fomentadas en todas ellas. En este trabajo se presenta una experiencia práctica del uso del Taller orientado para que el alumnado adquiera algunas de las competencias transversales de la asignatura de Construcciones II-EPSEB. En él, se alienta a los alumnos a trabajan aspectos de redacción técnica y de su técnica de aprendizaje extra-clase, además de también evalúa los trabajos de otros compañeros aplicando rúbricas preestablecidas con criterios de evaluación mixto (de conocimiento y de competencia trasversal).Peer Reviewe